In Figure REF , we display the series of the out-of-sample forecasts of VaR and ES, for each of the three stock indices, estimated by assuming the selected CAViaR-AS specification in both the univariate and the joint approaches. VaR predictions obtained with the univariate approach of [1]} are represented by the dotted blue line, while the VaR estimates produced by our joint approach are depicted in the solid red line. The estimated ES is represented by the dotted green line (using the univariate approach) and the solid orange line (using the joint approach). Left panels of Figure REF  refer to \(\tau  = [0.1, 0.1, 0.1]\) , the center panels report the case \(\tau  = [0.05, 0.05, 0.05]\)  while the right ones consider \(\tau  = [0.01, 0.01, 0.01]\) . The gray dots denote the original return series of each stock index. In all the cases, the estimates of VaR and ES produced by our joint approach lie below the corresponding values obtained with the univariate setting, suggesting that our proposed method could lead to more conservative results.
